Equality k(X) = ψ_c(X) for Urysohn spaces with an H-closed π-base

Determine whether, for every Urysohn Hausdorff space X that admits a π-base whose elements have H-closed closures, the cardinal function k(X) equals the closed pseudocharacter ψ_c(X).

Background

The paper notes that for H-closed spaces one has χ(X_s) ≤ ψ_c(X_s), and consequently k(X) = χ(X_s) = ψ_c(X). This raises the explicit question of whether the same equality k(X) = ψ_c(X) extends to the broader class of Urysohn spaces that merely have an H-closed π-base.